Ebook: Analysis and Geometry on Complex Homogeneous Domains

A number of important topics in complex analysis and geometry are covered in this excellent introductory text. Written by experts in the subject, each chapter unfolds from the basics to the more complex. The exposition is rapid-paced and efficient, without compromising proofs and examples that enable the reader to grasp the essentials. The most basic type of domain examined is the bounded symmetric domain, originally described and classified by Cartan and Harish- Chandra. Two of the five parts of the text deal with these domains: one introduces the subject through the theory of semisimple Lie algebras (Koranyi), and the other through Jordan algebras and triple systems (Roos). Larger classes of domains and spaces are furnished by the pseudo-Hermitian symmetric spaces and related R-spaces. These classes are covered via a study of their geometry and a presentation and classification of their Lie algebraic theory (Kaneyuki). In the fourth part of the book, the heat kernels of the symmetric spaces belonging to the classical Lie groups are determined (Lu). Explicit computations are made for each case, giving precise results and complementing the more abstract and general methods presented. Also explored are recent developments in the field, in particular, the study of complex semigroups which generalize complex tube domains and function spaces on them (Faraut). This volume will be useful as a graduate text for students of Lie group theory with connections to complex analysis, or as a self-study resource for newcomers to the field. Readers will reach the frontiers of the subject in a considerably shorter time than with existing texts.

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A number of important topics in complex analysis and geometry are covered in this excellent introductory text. Written by experts in the subject, each chapter unfolds from the basics to the more complex. The exposition is rapid-paced and efficient, without compromising proofs and examples that enable the reader to grasp the essentials. The most basic type of domain examined is the bounded symmetric domain, originally described and classified by Cartan and Harish- Chandra. Two of the five parts of the text deal with these domains: one introduces the subject through the theory of semisimple Lie algebras (Koranyi), and the other through Jordan algebras and triple systems (Roos). Larger classes of domains and spaces are furnished by the pseudo-Hermitian symmetric spaces and related R-spaces. These classes are covered via a study of their geometry and a presentation and classification of their Lie algebraic theory (Kaneyuki). In the fourth part of the book, the heat kernels of the symmetric spaces belonging to the classical Lie groups are determined (Lu). Explicit computations are made for each case, giving precise results and complementing the more abstract and general methods presented. Also explored are recent developments in the field, in particular, the study of complex semigroups which generalize complex tube domains and function spaces on them (Faraut). This volume will be useful as a graduate text for students of Lie group theory with connections to complex analysis, or as a self-study resource for newcomers to the field. Readers will reach the frontiers of the subject in a considerably shorter time than with existing texts.

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A number of important topics in complex analysis and geometry are covered in this excellent introductory text. Written by experts in the subject, each chapter unfolds from the basics to the more complex. The exposition is rapid-paced and efficient, without compromising proofs and examples that enable the reader to grasp the essentials. The most basic type of domain examined is the bounded symmetric domain, originally described and classified by Cartan and Harish- Chandra. Two of the five parts of the text deal with these domains: one introduces the subject through the theory of semisimple Lie algebras (Koranyi), and the other through Jordan algebras and triple systems (Roos). Larger classes of domains and spaces are furnished by the pseudo-Hermitian symmetric spaces and related R-spaces. These classes are covered via a study of their geometry and a presentation and classification of their Lie algebraic theory (Kaneyuki). In the fourth part of the book, the heat kernels of the symmetric spaces belonging to the classical Lie groups are determined (Lu). Explicit computations are made for each case, giving precise results and complementing the more abstract and general methods presented. Also explored are recent developments in the field, in particular, the study of complex semigroups which generalize complex tube domains and function spaces on them (Faraut). This volume will be useful as a graduate text for students of Lie group theory with connections to complex analysis, or as a self-study resource for newcomers to the field. Readers will reach the frontiers of the subject in a considerably shorter time than with existing texts.
Content:
Front Matter....Pages i-xvii
Front Matter....Pages 1-1
Introduction....Pages 3-4
Hilbert Spaces of Holomorphic Functions....Pages 5-18
Invariant Cones in a Lie Algebra and Complex Semi-groups....Pages 19-32
Positive Unitary Representations....Pages 33-43
Hilbert Function Spaces on Complex Semi-groups....Pages 45-63
Hilbert Function Spaces on a Complex Olshanski Semi-group in a Complex Semi-simple Lie Group....Pages 65-81
Front Matter....Pages 83-102
Introduction....Pages 103-103
Semisimple Graded Lie Algebras....Pages 105-106
Symmetric R-Spaces....Pages 107-126
Pseudo-Hermitian Symmetric Spaces....Pages 127-150
Front Matter....Pages 151-182
Introduction....Pages 183-183
Bergman Kernel and Bergman Metric....Pages 185-186
Symmetric Domains and Symmetric Spaces....Pages 187-191
Construction of the Hermitian Symmetric Spaces....Pages 193-202
Structure of Symmetric Domains....Pages 203-210
The Weighted Bergman Spaces....Pages 211-213
Differential Operators....Pages 225-242
Function Spaces....Pages 243-255
Front Matter....Pages 257-281
Introduction....Pages 283-283
Front Matter....Pages 285-302
The Laplace-Beltrami Operator in Various Coordinates....Pages 283-283
The Integral Transformations....Pages 303-320
The Harmonic Forms on the Complex Grassmann Manifold....Pages 321-336
The Horo-hypercircle Coordinate of a Complex Hyperball....Pages 337-350
The Matrix Representation of NIRGSS....Pages 351-363
Front Matter....Pages 365-380
Introduction....Pages 381-391
Polynomial Identities....Pages 393-424
Jordan Algebras....Pages 425-425
The Quasi-inverse....Pages 427-428
The Generic Minimal Polynomial....Pages 429-449
Tripotents and Peirce Decomposition....Pages 451-460
Hermitian Positive JTS....Pages 461-473
Further Results and Open Problems....Pages 475-496
Back Matter....Pages 497-505
....Pages 507-527